Finding average velocity in uniform circular motion requires understanding that velocity is a vector quantity (having both magnitude and direction) and differs significantly from speed (the magnitude of velocity). Average velocity is calculated as the total displacement divided by the total time taken.
Understanding Velocity vs. Speed in Circular Motion
In uniform circular motion, an object moves in a circular path at a constant speed. While the speed remains constant, the direction of the velocity vector is continuously changing as the object follows the curve of the circle. This means the velocity itself is not constant, even though the speed is.
Instantaneous Speed (Circular Velocity) - As Per Reference
The provided reference, "Circular Velocity Formula FAQs," states: "Velocity in circular motion problem is found by using the formula for circular velocity which are given by Vc=2πrT or Vc=ωr." Note: The formula Vc=2πrT is likely a typo in the source; the correct formula for speed (Vc) or instantaneous velocity magnitude is typically Vc=2πr/T, where T is the period.
These formulas, Vc=2πr/T (using the corrected version assuming T is the period) and Vc=ωr, describe the constant speed of the object along the circular path at any given instant.
Here's a quick look at the terms:
| Term | Symbol | Meaning | Unit (SI) |
|---|---|---|---|
| Speed | Vc | Magnitude of instantaneous velocity | m/s |
| Radius | r | Radius of the circular path | m |
| Period | T | Time for one complete revolution | s |
| Angular Speed | ω (omega) | Rate of change of angular position | rad/s |
These formulas allow you to calculate how fast the object is moving, but they do not directly give you the average velocity over a time interval, because average velocity considers the displacement.
Calculating Average Velocity
Average velocity ($\vec{v}_{avg}$) is defined as the total displacement ($\vec{\Delta r}$) divided by the time interval ($\Delta t$):
$\vec{v}_{avg} = \frac{\vec{\Delta r}}{\Delta t}$
Displacement is a vector pointing from the initial position to the final position. In circular motion, the displacement depends entirely on the starting and ending points over the specified time interval.
Average Velocity Over a Full Cycle
If the time interval ($\Delta t$) is exactly one period (T), the object completes one full revolution. It starts and ends at the same point.
- Displacement ($\vec{\Delta r}$) = 0 (since the final position is the same as the initial position).
- Time interval ($\Delta t$) = T.
Therefore, the average velocity over a full cycle is:
$\vec{v}_{avg} = \frac{0}{T} = 0$
The average velocity over any complete number of cycles is always zero.
Average Velocity Over a Partial Cycle
For a time interval less than a full cycle, the displacement is the straight-line distance and direction from the starting point to the ending point, represented by a chord of the circle.
- Displacement ($\vec{\Delta r}$): This is a vector quantity whose magnitude is the length of the chord connecting the initial and final positions, and whose direction points along that chord. Calculating this requires knowing the angle traversed during the time interval $\Delta t$. The angle $\theta$ (in radians) is $\theta = \omega \Delta t = (2\pi/T) \Delta t$. The magnitude of the displacement vector is $2r \sin(\theta/2)$.
- Time interval ($\Delta t$).
The magnitude of the average velocity is $\frac{2r \sin(\theta/2)}{\Delta t}$, and its direction is the direction of the chord.
Examples for Specific Partial Cycles:
-
Half a Cycle (Time $\Delta t = T/2$)
- The object moves from one side of the circle to the exact opposite side.
- Displacement ($\vec{\Delta r}$) = a vector across the diameter. Magnitude = 2r.
- Time interval ($\Delta t$) = T/2.
- Magnitude of Average Velocity = $\frac{2r}{T/2} = \frac{4r}{T}$.
- Using $Vc = 2\pi r/T$, we know $1/T = Vc/(2\pi r)$. So, magnitude = $4r \cdot \frac{Vc}{2\pi r} = \frac{2Vc}{\pi}$. The direction is along the diameter connecting the start and end points.
-
Quarter Cycle (Time $\Delta t = T/4$)
- The object moves through a 90-degree angle ($\pi/2$ radians).
- Displacement ($\vec{\Delta r}$) = the hypotenuse of a right triangle with two sides of length r. Magnitude = $\sqrt{r^2 + r^2} = r\sqrt{2}$.
- Time interval ($\Delta t$) = T/4.
- Magnitude of Average Velocity = $\frac{r\sqrt{2}}{T/4} = \frac{4r\sqrt{2}}{T}$.
- Using $Vc = 2\pi r/T$, magnitude = $\frac{4r\sqrt{2}}{T} = \sqrt{2} \cdot \frac{4r}{T} = \sqrt{2} \cdot \frac{2}{\pi} \cdot \frac{2\pi r}{T} = \frac{2\sqrt{2}}{\pi} Vc$. The direction is along the chord connecting the start and end points, at a 45-degree angle to the radii at the start and end points.
In summary, while formulas like Vc = 2πr/T or Vc = ωr describe the constant instantaneous speed in uniform circular motion, finding the average velocity requires calculating the vector displacement over the specific time interval and dividing by that interval.
